Optimization of national grain imports to balance risk and return: a portfolio theory approach

The portfolio approach was developed for application in finance to assess the risk and return of asset investing (Markowitz 1952). It is a flexible decision-making tool that seeks to either maximize the expected return or minimize the collective risk among alternative portfolios of assets (Elton and Gruber 1997). Portfolio theory encourages individual decision-makers to diversify their investments, such that 'all eggs are not put in a single basket' (Fraser et al 2005). Here, we use modern portfolio theory to assess the risk vs expected return for grain trade. Specifically, we focus on grain imports since these grain supplies are a key part of domestic food security (Donaldson 1984).

In this study, each importer nation represents a portfolio manager and every exporter nation is an available asset that could be invested in. Here, the fraction of investment in each asset is represented by import mass share. The return of each asset is defined to be the weighted average unit value of import ($ kg⁻¹). The unit value of import is also the cost paid by the portfolio manager. For grain imports, a major goal of the portfolio manager is thus to minimize return (i.e. cost). This makes the application of modern portfolio theory to grain trade distinct from asset investing, which aims to maximize return (i.e. profit). We consider both cross-sectional and panel risk. The portfolio cross-sectional risk is defined to be the variance in unit value ($ kg⁻¹) across the invested assets (i.e. exporters) in a single time period (Bollino and Galkin 2021). The portfolio panel risk is the collective covariance in unit value ($ kg⁻¹) across the invested assets (i.e. exporters) through time (Markowitz 1956). Thus, cross-sectional risk is used in the trade-off analysis between risk and return, whereas panel risk is used in the mean-variance optimization model.

A mean-variance optimization model was developed to identify less risky grain import portfolios with and without cost considerations. To optimize a portfolio, the manager aims to minimize the collective volatility across trade partners (i.e. assets) through time. The minimum covariance helps asset managers (i.e. importers) invest in assets whose change in unit value ($ kg⁻¹) is not positively correlated through time. The portfolio manager optimizes by investing in assets that both increase and decrease in cost at the same time to stabilize its return. This approach means that grain importers minimize the cost covariance across their assets (i.e. trade partners) to reduce uncertainty in unit value. The optimization of national grain import portfolios allocates trade to exporters with available mass while minimizing the volatility in import cost. A supply mass constraint is used to ensure that exporters have the ability to meet the optimized portfolio of each importer nation one at a time.

We use annual 'import' reports from COMTRADE (UNCOMTRADE 2020) both in value ($) and mass (kg) that are reported by the importer nations for the years 2008–2020. Trade data is organized as a square matrix with exporters in the rows and importers in the columns. Separate matrices are constructed for the wheat, maize, and rice trade in units of mass (kg) and value ($). Then, we divide the total import value ($) by the total import mass (kg) between every exporter and importer nation to obtain the 'unit value ($ kg⁻¹) of the commodity per trade relationship'. Our cost measure represents the Cost-Insurance-Freight of a commodity that is experienced by the importer nation which includes insurance and transportation (FAOSTAT 2020, Chen et al 2022). Hence, the import unit cost is generally higher than the corresponding export unit cost.

Data for the year 2020 serves as the most up-to-date empirical case to evaluate the trade-off between cross-sectional risk and return in grain import portfolios. Data between 2008–2019 is used to calculate the mean and covariance in unit value of import ($ kg⁻¹) through time for determining the optimal and optimum portfolios.

Here, we explain how we apply the modern portfolio theory concepts of expected return, risk, and diversity to grain trade. For a single time period, the expected return of grain imports is defined to be the weighted average in the unit value ($ kg⁻¹) of each commodity. The cross-sectional risk of the grain import portfolio is given by the variance of its return (i.e. cost) among the assets (i.e. exporters) (Bollino and Galkin 2021). Expected return E_j and cross-sectional risk R_j of the importer j are given in equations (1) and (2), respectively. The mass share of imports provided by each exporter i is represented by w_i , and c_i is the unit value of the commodity ($ kg⁻¹) charged by exporter i to the importer j (see SI for more detail).

$$\begin{equation} E_j = \sum_{i = 1}^{N} w_{i}\times c_{i} \end{equation} \tag{ 1 }$$

$$\begin{equation} R_j = {\mathrm {var}}\left(w_{i}\times c_{i}\right) ~ i = 1,\ldots,N \end{equation} \tag{ 2 }$$

The Herfindahl–Hirschman Index (HHI) is used to calculate the portfolio diversity of each importer j and is given in equation (3). Lower HHI values indicate the mass shares are more evenly distributed among a greater number of suppliers, hence more diverse import portfolios (Rhoades 1993).

$$\begin{equation} HHI_j = \sum_{i = 1}^{N} w_{i}^2 \end{equation} \tag{ 3 }$$

We first assess the relationship between expected return E_j and cross-sectional risk R_j in the grain portfolios of each importer j for a single year. We use 2020 empirical trade data to compute E_j and R_j . Modern portfolio theory encourages a diverse set of investments to reduce risk. In our case, this corresponds to importing from a diverse set of grain suppliers. Thus, we also assess the relationship between diversity HHI_j and cross-sectional risk R_j in grain portfolios of importer nations for the year 2020. These analyses lay the foundation for the mean-variance optimization model approach, and the observed risk vs. return trends in grain imports motivate the proposal of optimal and optimum portfolios.

To identify optimal and optimum portfolios, we develop a mean-variance optimization model with an exporter mass supply constraint as in equations (4)–(8). The mean-variance optimization is a data-driven model which requires historic trends in the unit cost to compute its mean and covariance (Markowitz 1956). So, trade data from 2008–2019 that is specific to each grain commodity and importer nation is used to compute the expected return and panel risk of potential portfolios. The optimization model is formulated to be run individually for each major grain importer j to propose alternative portfolios for the year 2020 based on historical data (see SI for more detailed information on the optimization model).

The decision that is being made by the model for each importer nation is: 'How much to import from which available exporters for the year 2020, to minimize the portfolio risk?' By design, generally, more diverse portfolios are proposed by mean-variance optimization model i.e. more even mass share distribution across multiple exporters to reduce the panel risk.

$$\begin{equation} \min_{w} \left[w\right]^T \times\left[cov\right]\times\left[w\right] \end{equation} \tag{ 4 }$$

$$\begin{equation} \textrm{s.t.} \sum_{i = 1}^{N} w_{i} = 1 \end{equation} \tag{ 5 }$$

$$\begin{equation} \sum_{i = 1}^{N} w_{i}\times c_{i} = E_j \end{equation} \tag{ 6 }$$

$$\begin{equation} D_j\times w_i \leq S_i~\forall i \in N \end{equation} \tag{ 7 }$$

$$\begin{equation} 1\unicode{x2A7E} w_{i}\unicode{x2A7E}0~\forall i \in N \end{equation} \tag{ 8 }$$

For each importer nation j, we run the mean-variance optimization model individually and propose two separate portfolios: optimal and optimum (i.e. proposals for 2020). For the optimal portfolio (i.e. with cost constraint), we run the presented version of the model. For the optimum portfolio (i.e. without any cost constraint), we run the presented model without the second constraint (equation (6)). Thus, optimal and empirical portfolios in 2020 per importer nation j have the exact same unit cost, E_j . However, optimum portfolios are not bounded by any specific cost.

The mass supply constraint (equation (7)) ensures that each exporter has the capacity to meet the portfolio requirements of the importer j, which we assume to be a small open economy. Policies in small open economies do not alter world prices, i.e. these nations take world prices as given, which is a common assumption in macroeconomics (Vegh 2013). Even though case study nations are small open economies, their total demand could be high enough that multiple exporters might be required to allocate supply. Thus, to enhance the realism of our approach, each exporter's capacity is bounded by its available export supply mass.

A summary of the key terms in our study and their definitions are listed in table 1.

The relationship between cross-sectional risk R_j and return E_j in grain import portfolios is shown in figure 1. There is a positive relationship between cross-sectional risk and return in 2020 for wheat, maize, and rice import portfolios, respectively. This positive relationship means that high-risk import portfolios also have a higher return, as in the application of portfolio theory to financial instruments. However, this also means that nations with more risk in their grain imports also pay more for those imports. This is because, unlike the application of portfolio theory to finance, here the return captures the cost paid by the importer. Thus, importing nations aim to minimize both risk and return/cost. However, our observation indicates that importer nations are not paying higher costs to achieve greater resilience. In fact, even though they are paying higher costs, they are subject to higher import volatility.

Zoom In Zoom Out Reset image size

Asset diversification is not the main objective of portfolio theory but is an important component as it relates to portfolio risk and reliability (see SI for more detailed information on portfolio reliability). The relationship between diversity HHI_j and cross-sectional risk R_j in grain imports is shown in figure 2. We observe that there exists a positive relationship between diversity (HHI_j ) and cross-sectional risk (R_j ). Note that higher values of HHI indicate lower levels of diversity, so the positive relationship indicates that risk increases as diversity decreases. This means that more diverse portfolios also provide lower volatility in the cost of import. If there was a shock in a supplier and its grain supply was lost, the importers with more diverse portfolios (lower y-axis values) would experience less change in their average unit cost of import (lower x-axis values). Thus, more diverse portfolios are also less risky as found in modern portfolio theory. This finding corroborates recent work that shows that increasing diversity in food supplies reduces supply chain vulnerability to risks (Gomez et al 2021).

Zoom In Zoom Out Reset image size

We also perform additional portfolio diversity analysis that considers the reliability of different exporters in terms of their governmental (WGI, 2020b), environmental (EPI, 2020), and logistical (LPI, 2020a) performance. The assessment of the exporters' reliability stems from the fact that nations do not have uniform supply chain attributes. Based on supplier selections and trade dependencies, importer nations could be subject to more disruptions across their international trading network. For example, some exporters tend to experience more internal conflicts, less adaptability against climate change, and not well-automated supply chains. Importer nations that rely on these less reliable suppliers are likely to be subject to more disruptions in their import supplies (see SI for a more detailed explanation of the reliability indices and their calculation).

In figure 3, the observed positive relationship between diversity and reliability indices-weighted diversity (Goddin 2019) of import portfolios is illustrated (see SI for a more detailed explanation of the reliability indices-weighted diversity). This positive relationship indicates that importers with less diverse portfolios (higher HHI values) also have less reliable suppliers (higher reliability indices-weighted HHI values), whereas more diverse import portfolios are composed of more reliable exporters. This is consistent across the governmental, environmental, and logistical performances of exporters. Thus, increasing diversity in supplier selections also improves the logistical, environmental, and governmental reliability of import portfolios. This relationship supplements previous literature and our findings on the benefits of import portfolio diversity against supply chain shocks (Gomez et al 2021).

Zoom In Zoom Out Reset image size

The observed positive trade-off between risk and return highlights the fact that importer nations do not achieve less risky portfolios by paying more for their grain imports. On the contrary, nations that pay more are subject to higher volatility in their imports in case of a disruption in one of their suppliers. This empirical conclusion leads us to consider whether importers could achieve less risky portfolios through adjustments in their supplier selections and proportions. Is it possible to uncover grain import portfolios with lower volatility and a more diverse set of suppliers, that are also economically attractive?

To answer this question, we develop a mean-variance optimization model and propose optimum and optimal portfolios for major grain importers individually. The case study nations are Egypt for wheat import, Vietnam for maize import, and Saudi Arabia for rice import. Note that these low-income and generally food-insecure nations are selected from the list of the top 10 importers of each grain commodity (see table 2). Through optimal and optimum portfolios, the model allocates how much to import from available exporters to minimize the panel risk for each nation in 2020.

Figure 4 compares the mass share distribution to the available suppliers for the empirical, optimal, and optimum portfolios. Empirical portfolios indicate the real-world import scenarios for the case study nations in 2020 (figures 4(A)–(D)–(G)). Optimal portfolios show minimum panel risk while keeping the import cost constant (figures 4(B)–(E)–(H)). Optimum portfolios for these nations minimize panel risk without cost constraint (figures 4(C)–(F)–(I)). This approach enables us to identify grain import portfolios that are less risky (and more diverse) than the realized import portfolios for the year 2020. Importantly, these case studies highlight opportunities to minimize panel risk at lower cost.

Zoom In Zoom Out Reset image size

The composition of total mass import (kg) across empirical, optimal, and optimum portfolios for the case study nations is shown in figure 5. The total amount of mass (kg) imported is the same across portfolios, but the composition across suppliers changes. The mass supplied by each exporter nation differs across empirical, optimal, and optimum portfolios. Notably, these portfolios are physically achievable due to the supplier mass constraint in the optimization model per importer nation (see section 2). Both optimal and optimum scenarios select the available suppliers that collectively have the lowest covariance through time (e.g. volatility in unit value is minimized). This means that these portfolios are composed of suppliers whose historic change in unit value moves in opposite directions to one another.

Zoom In Zoom Out Reset image size

More detailed information on the panel risk, diversity, and cost across scenarios for the case study nations is provided in table 3. Egypt's optimal wheat portfolio reduces panel risk by 54% while also increasing diversity by 21% and keeping cost constant. More importantly, Egypt's optimum wheat portfolio reduces volatility by 98%, increases diversity by 39%, and also reduces cost by 14%. Similarly, Vietnam's optimal maize portfolio achieves 93% lower panel risk and 1.5% higher diversity than real-world trade without costing more. Vietnam's optimum maize portfolio achieves less panel risk (93% decrease) and more diversity (1.6% increase) at a lower cost (1.4% reduction). Saudi Arabia's optimal rice portfolio is 78% less risky and 70% more diverse with constant cost. Again, the optimum portfolio reduces panel risk (88% lower) and increases diversity (73% higher) at a lower cost (23% lower). These results identify that the case study nations may be able to achieve less risky and more diverse grain import portfolios at the same or lower cost. These case study findings are consistent with global results (see figure 2) in which more diverse portfolios are also less risky.

Table 3. Key portfolio optimization characteristics for the major grain importers considered in this study (rounded to 3-decimal points). The mean unit value ($ kg⁻¹) matches between the empirical and optimal portfolios by definition.

Egypt's Wheat Imports
	Empirical portfolio	Optimal portfolio	Optimum portfolio
Panel risk	$2.54 \times 10^{-3}$	$1.16 \times 10^{-3}$	$5.93 \times 10^{-5}$
Unit cost ($ kg−1)	0.281	0.281	0.242
Diversity	0.436	0.346	0.266
Vietnam's Maize Imports
	Empirical portfolio	Optimal portfolio	Optimum portfolio
Panel risk	$9.51 \times 10^{-4}$	$6.67 \times 10^{-5}$	$6.66 \times 10^{-5}$
Unit cost ($ kg−1)	0.198	0.198	0.195
Diversity	0.499	0.492	0.491
Saudi Arabia's Rice Imports
	Empirical portfolio	Optimal portfolio	Optimum portfolio
Panel risk	$1.78 \times 10^{-2}$	$3.84 \times 10^{-3}$	$2.19 \times 10^{-3}$
Unit cost ($ kg−1)	0.913	0.913	0.700
Diversity	0.651	0.196	0.175

In our framework, the mass supply constraint is introduced for each exporter but does not serve as a global balance. Thus, this framework is not meant to be applied to all world nations simultaneously, which would require a general equilibrium framework in which global prices would adjust with varying demands. Instead, portfolio theory is a suitable framework for small, open economies, whose portfolio adjustments would not significantly impact world prices. However, the dual objectives of grain supply affordability and stability are the most important in small, open economies, which makes our application of portfolio theory to this context suitable. Therefore, we provide individually tailored import strategies based on the historic trade relationships of these nations. Our mean-variance optimization approach could be a useful preparedness tool against potential supply disruptions, especially for low-income and generally food-insecure importers.

It is also important to note that we are not able to identify optimal or optimum portfolios that improve upon the empirical cases across all three objectives (i.e. risk, cost, and diversity of each portfolio) for all countries. This means that certain countries cannot reduce panel risk in their grain imports for the same or lower cost of imports. In other words, to reduce the riskiness of their grain import portfolios, some countries will need to spend more. This is due to each nation's grain import priorities and historic suppliers. For example, Turkey would need to spend 26% more money on wheat imports to achieve 86% more stable supplies, as seen in table 4 (also see figure 6 for Turkey's mass share distribution across available suppliers for the empirical, optimal, and optimum portfolios). This highlights that optimal and optimum scenarios with better diversity, cost, and stability can be identified for some but not all importers. However, this approach can help countries calculate how much diversifying their grain imports to reduce panel risk is likely to cost.

Zoom In Zoom Out Reset image size

Table 4. Key portfolio optimization characteristics for other grain importers considered in this study (rounded to 3-decimal points). Turkey needs to pay more to achieve a more stable wheat import portfolio. The mean unit value ($ kg⁻¹) matches between the empirical and optimal portfolios by definition.

Turkey's Wheat Imports
	Empirical portfolio	Optimal portfolio	Optimum portfolio
Panel risk	$7.68 \times 10^{-4}$	$2.16 \times 10^{-4}$	$1.09 \times 10^{-4}$
Unit cost ($ kg−1)	0.242	0.242	0.306
Diversity	0.485	0.358	0.278

This distinction on Turkey's wheat imports is especially important when the positive trade-off between risk and return is considered. Due to this positive relationship, decision-makers might assume that always choosing the lowest available cost would ensure the minimum risk scenario. Further, they might switch to the next lowest cost in case a disruption occurs in their first choice supplier. However, minimum-cost portfolios do not always provide the minimum risk. If minimum cost is the priority, a new optimization model with an objective function that aims to minimize the total cost of import must be formulated and solved.

Lastly, it is important to highlight that the mean-variance optimization approach might miss some real-world details as it is based on only historical data, like other empirical approaches. In reality, altering trade relationships could involve additional costs and other socio-political issues among nations, which might make it difficult to build new trade relationships. In addition, an a-historical shock (i.e. the recent invasion of Ukraine) would not be represented in the model in its original form. For example, the optimal and optimum maize import portfolios of Vietnam suggest more reliance on both Ukraine and Russia, which has proven to be risky in light of the recent conflict. However, the mean-variance optimization is an adaptable approach and these concerns could be addressed through further modifications in the model. Through the inspiration driven by our reliability analysis of grain imports (see SI for more detail), more reliable exporter nations could be prioritized during the supplier selection, which we suggest as an important avenue for future research. Thus, a mean-variance optimization framework would be able to identify opportunities to reduce the risk of grain import portfolios (for the same cost or less) while taking other practical factors into account.

We demonstrate that portfolio theory is applicable to grain imports, which highlights the important trade-offs between risk, cost, and diversity that policy-makers may want to consider. As an example, the recent Russian invasion of Ukraine highlighted that several countries were reliant on wheat supplies from the Black Sea region (Behnassi and El Haiba 2022). Wheat from the Black Sea region is usually cheaper than wheat supplies from other regions, such as Europe or the USA (Abay et al 2023), which likely explains why several countries relied on Ukraine for their wheat imports. However, our findings suggest that major importers should diversify their sourcing so that they are better prepared for future crises. Importantly, our findings highlight that it may be possible for grain importers to increase their supplier diversity while also keeping costs constant (e.g. see the case of Egyptian wheat imports in section 3.3).

Another policy implication of our study is that trade can buffer shocks when the portfolio selection explicitly considers risk. For this reason, risk should be included in the policy mix to enhance the food security of importing nations. Importing nations can select their grain suppliers and the quantity to import from each supplier to minimize risk with the portfolio framework that we developed in this study. This approach will penalize countries that have historically implemented trade barriers during crises since these bans would have increased the volatility of those exporter's prices. Policy-makers often aim to discourage trade barriers during crises (Anderson 2022), since they increase the price of global supplies and tend to be contagious (Mottaleb et al 2022, Abay et al 2023). The mean-variance optimization framework penalizes countries that implement export bans by making them less desirable for importers seeking price stability. Importantly, the mean-variance optimization framework is flexible and could be adapted to include other national characteristics that policy-makers are interested in prioritizing, e.g. such as prioritizing trade with other democracies, countries in a free-trade zone, nations with high-quality infrastructure, etc.

The policy implications of our study primarily relate to trade policies, which are a complement but not a substitute for domestic food security policies (Dithmer and Abdulai 2017, Spiker et al 2023). Nations that are vulnerable to food price spikes may want to consider creating safety nets, as these can be effective tools to support themselves against potential food supply crises. An illustrative case is in Egypt where the government is currently expanding its existing socio-economic protection programs to shield their most vulnerable citizens against food insecurity (Belhaj et al 2022). However, our study is motivated by the fact that global grain markets tend to be more stable than domestic markets since they are more diverse, and global production has more adaptive capacity to withstand shocks than does any individual country (Lampietti et al 2011). For this reason, our study focuses on opportunities to enhance food security through the trade system, which means that domestic food security policies are outside the scope.

This study enables importer nations to determine how far their current trade portfolio is from an optimal and optimum benchmark. One of the main advantages of this benchmarking approach may be for helping decision-makers determine how to reduce risk in their grain imports in a realistic way, due to the mass supply constraint on exporters and cost considerations. By comparing empirical and optimal portfolios, each nation can evaluate opportunities to adjust their trade relationships and policies to achieve less risky import portfolios at the same average unit cost. Further, importer nations can use our approach to identify if they can achieve both less risky and less costly imports, which are also more diverse, by comparing their empirical and optimum portfolios.

The main limitation of using modern portfolio theory in the context of grain trade is the lack of high-frequency historical data, when compared with financial settings. In a typical modern portfolio theory application, daily data on stock prices for a period of decades could be used to propose future investment opportunities. Unfortunately, this granularity in time records is not available for the trade statistics provided by UNCOMTRADE. Bilateral trade data is limited to the annual time step. Additionally, there are known uncertainties in the country reports that underpin the UNCOMTRADE database (refer to Chen et al 2022 for a discussion of outliers, missing values, and bilateral asymmetries in the UNCOMTRADE data). Uncertainties in UNCOMTRADE have primarily been noted for the material flow of goods, but we also utilize the financial value information in this study, which has less uncertainty associated with it, and, coupled with the material flow data lends credibility to our approach.

Despite these limitations, the UNCOMTRADE database is one of the most widely used and accepted databases of international trade across disciplines (Chen et al 2022). Thus, our use of the UNCOMTRADE database at the annual time scale is consistent with the literature, but novel in the application of modern portfolio theory to national grain imports accounts.